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Theorem lei3 246
 Description: L.e. to Kalmbach implication.
Hypothesis
Ref Expression
lei3.1 ab
Assertion
Ref Expression
lei3 (a3 b) = 1

Proof of Theorem lei3
StepHypRef Expression
1 ax-a3 32 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = ((ab) ∪ ((ab ) ∪ (a ∩ (ab))))
2 ax-a2 31 . . . . 5 (ba) = (ab )
3 ancom 74 . . . . . . 7 (ab ) = (ba )
4 lei3.1 . . . . . . . . 9 ab
54lecon 154 . . . . . . . 8 ba
65df2le2 136 . . . . . . 7 (ba ) = b
73, 6ax-r2 36 . . . . . 6 (ab ) = b
84sklem 230 . . . . . . . 8 (ab) = 1
98lan 77 . . . . . . 7 (a ∩ (ab)) = (a ∩ 1)
10 an1 106 . . . . . . 7 (a ∩ 1) = a
119, 10ax-r2 36 . . . . . 6 (a ∩ (ab)) = a
127, 112or 72 . . . . 5 ((ab ) ∪ (a ∩ (ab))) = (ba)
13 anor2 89 . . . . . 6 (ab) = (ab )
1413con2 67 . . . . 5 (ab) = (ab )
152, 12, 143tr1 63 . . . 4 ((ab ) ∪ (a ∩ (ab))) = (ab)
1615lor 70 . . 3 ((ab) ∪ ((ab ) ∪ (a ∩ (ab)))) = ((ab) ∪ (ab) )
171, 16ax-r2 36 . 2 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = ((ab) ∪ (ab) )
18 df-i3 46 . 2 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
19 df-t 41 . 2 1 = ((ab) ∪ (ab) )
2017, 18, 193tr1 63 1 (a3 b) = 1
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →3 wi3 14 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131 This theorem is referenced by:  bina3  284  bina4  285  bina5  286  bii3  516  binr1  517  binr2  518  binr3  519  i3ri3  538  i3li3  539  i32i3  540  i3th5  547  i3th7  549  i3th8  550
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